document.write( "Question 1199250: The half-life of a radioactive substance is one hundred twenty-five days. How many days will it take for eighty-two percent of the substance to decay?\r
\n" ); document.write( "\n" ); document.write( "A. 180
\n" ); document.write( "B. 367
\n" ); document.write( "C. 275
\n" ); document.write( "D. 310
\n" ); document.write( "E. 532
\n" ); document.write( "F. 426 \n" ); document.write( "
Algebra.Com's Answer #833040 by ikleyn(52803)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "The half-life of a radioactive substance is one hundred twenty-five days. How many days will it take for eighty-two percent of the substance to decay?
\n" ); document.write( "A. 180
\n" ); document.write( "B. 367
\n" ); document.write( "C. 275
\n" ); document.write( "D. 310
\n" ); document.write( "E. 532
\n" ); document.write( "F. 426
\n" ); document.write( "~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "The half-life is given, so I write the exponential decay function \r\n" );
document.write( "\r\n" );
document.write( "in the base  \"1%2F2\"  in terms of half-lives  (in a way as it \"highlight%28highlight%28SHOULD%29%29\" be done based on given info)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    m(t) = \"m%5B0%5D%2A%281%2F2%29%5E%28t%2F125%29\",    (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "where m(t) is the current remaining mass, \"m%5B0%5D\" is the starting mass and t is the time, in days.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We are given that m(t) = \"%281-0.82%29%2Am%5B0%5D\" = \"0.18%2Am%5B0%5D\", so  \"m%28t%29%2Fm%5B0%5D\" = 0.18.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Therefore, equation (1) takes the form\r\n" );
document.write( "\r\n" );
document.write( "    \"%281%2F2%29%5E%28t%2F125%29\" = 0.18.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We want to find \"t\" from this equation.  For it, transform it step by step.\r\n" );
document.write( "First step is to take logarithm base 10 of both sides\r\n" );
document.write( "\r\n" );
document.write( "    \"%28t%2F125%29%2Alog%28%280.5%29%29\" = log(0.18)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From it, express t and calculate\r\n" );
document.write( "\r\n" );
document.write( "    t = \"125%2A%28log%28%280.18%29%29%2Flog%28%280.5%29%29%29\" = 309.24 days.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "309.24 days is the required \"continuous\" time (real value).\r\n" );
document.write( "If you want to get the answer as an integer number of days with a necessary margin, \r\n" );
document.write( "round it to the closest greater integer number, which is 310.\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-----------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

Post-solution note :

\r
\n" ); document.write( "\n" ); document.write( "        when the half-life is given, write the exponential decay function in terms of half-life\r
\n" ); document.write( "\n" ); document.write( "        with the base   \"1%2F2\".   It is the commonly accepted traditional   GOOD  STYLE.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Writing in other form  (ekt-form),  as Edwin makes it,   IS  NOT  a good style when half-life is given.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Writing equation in the right form is not only the choice of the style.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Doing this way as I did,  you perform only those calculations,
\n" ); document.write( "that are  REALLY  necessary,  and do not make  UNNECESSARY  calculations.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "It is the major reason of choosing that or the other style/approach.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "///////////////\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "On radioactive decay,  see the lesson\r
\n" ); document.write( "\n" ); document.write( "    - Radioactive decay problems \r
\n" ); document.write( "\n" ); document.write( "in this site.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "You will find many similar  (and different)  solved problems there.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        Use this lesson as your handbook,  textbook,  guide,  tutorials, and  (free of charge)  home teacher.\r
\n" ); document.write( "\n" ); document.write( "        Learn the subject from there once and for all.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in  ALGEBRA-I  in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The referred lesson is the part of this online textbook under the topic \"Logarithms\".\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I
\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );